Correct Answer: encourage

Solution : The correct choice is the third option.

Explanation : The idiom egg on means to encourage or urge someone to do something, often implying pushing or provoking them to take a specific action. In the detailed grammatical explanation, it's essential to emphasise the idiomatic nature of the expression and how it conveys a sense of incitement or stimulation.

Therefore, the correct word that expresses the meaning of the idiom is encourage .